Holomorphic flat projective structures on projective threefolds
The author intends to show the usefulness of the recent methods of algebraic geometry (Mori theory) to the study of differential geometric structures on algebraic manifolds. Main Theorem: Let \(X\) be a smooth projective threefold with a holomorphic flat projective structure (i.e. a normal integrable projective connection) and Kodaira dimension \(\kappa \neq 1\), 2. Then: (a) \(\kappa<0 \Leftrightarrow X \cong \mathbb{P}^3\); (b) \(\kappa = 0 \Leftrightarrow\) there is a finite unramified covering of \(X\) by a torus; (c) \(\kappa = 3 \Leftrightarrow\) the universal covering of \(X\) is the unit ball of \(\mathbb{C}^3\). Remark: Let \(X\) be (a) a four-dimensional projective manifold with \(\kappa \geq 0\) and \(K_X\) not nef, or (b) a compact complex 3-fold in the class \(C\) with \(\kappa < 0\) and a nontrivial Albanese torus. Then \(X\) has no holomorphic flat projective structure. The main use of Mori theory is to show that suitable projective varieties contain a rational curve and to deform it inside the ambient projective variety.
scientific article; zbMATH DE number 792875